Gmail messages from theself aren’t ‘Important’

Your domain needs a Sender Policy Framework - see SPF record.

Though it is not a standard, it is highly recommended as an anti-SPAM measure.

It is optional but some stupid mail servers reject mail if this record is not present.

Here is Google Support on how to set it.

A strength of Gmail is the use of labels, instead of folders. You need to create a filter to automatically add a label to the message and automaticlly archive the messages.

To create a filer for a message that frequently is sent to you:

1. select or open a message you want to filter
2. Under the More pulldown select the command: Filter messages like these
3. You will then see the form open up with the FROM box filled in.
4. Google with then show you the messages that the filter has found.
5. Click create filter with this search.
6. You can now specify if you want to:
   • Skip the Inbox (Archive it)
   • Apply the label:
   • Also apply filter to x matching conversations.
7. Click create the filter.

Archiving the messages keeps the message, it just removes the inbox label from the message. Labels can be used to search using label:mylabel. Labels are listed on the sidebar menu so that you can quickly jump to a filter that just shows the messages with that label. Plus messages can have multiple labels.

You can edit/delete/create filters under Settings(gear)->Filter